So, equivalently polynomial `p(x)=x^3+3x^2-2x+1` can be written as `p(x)=(x-2)^3+9(x-2)^2+22(x-2)+17`.

Thus, Taylor formula for polynomials allows us to rewrite any polynomial in terms of `(x-a)`.

Now, let's see how we can use this idea for any differentiable functions.

Suppose that function `y=f(x)` has finite derivatives up to n-th order at point `a`.

Taylor Formula for any Function. For function `y=f(x)` n-th degree Taylor polynomial at point `x=a` is ` T_n(x)=f(a)+(f'(a))/(1!)(x-a)+(f''(a))/(2!)(x-a)^2+...+(f^((n-1))(a))/((n-1)!)(x-a)^(n-1)+(f^((n))(a))/(n!)(x-a)^n`.

Of course, `T_n(x)!=f(x)`, but as appeared `T_n(x)` is a very good approximation for `f(x)` when `x->a`. And the higher `n` (order of polynomial) the better approximation.

Fact. `T_n(x)~~f(x)` as `x->a`.

This fact allows us to approximate function by polynomial near point `x=a` with any precision we want, by taking high degree polynomial.